This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Phys. 7268 Assignment 2 Due: 2/10/11 Problem 1 Type of Linear Instability for a SwiftHohenberg Equation with a SecondOrder Time Derivative. Consider a twodimensional SwiftHohenberg equation with a secondorder time derivative: ∂ 2 t u ( x,y,t ) = ru (1 + ∇ 2 ) 2 u u 3 . Show that the base solution u = 0 becomes linearly unstable at r = 0 and determine the type of instability ( I s , I o , III o , etc). Problem 2 Type of Instability for Two Coupled ReactionDiffusion Equations. Consider the following reactiondiffusion evolution equations ∂ t u = au + ∂ 2 x u b∂ 2 x v ( u 2 + v 2 )( u + cv ) , ∂ t v = av + b∂ 2 x u + ∂ 2 x v ( u 2 + v 2 )( v cu ) for the scalar fields u ( x,t ) and v ( x,t ) in one dimension, where the parameters a , b , and c are arbitrary real constants. (You can think about the fields u and v as representing concentrations of two reacting and diffusing chemicals.) Derive a condition for the linear instability of the zero base solution ( u ,v ) = (0 , 0) and determine whether the instability is of type I s , I o , or III o . Hint: When you substitute the infinitesimal perturbation with the correct spatial and temporal dependence (the same for both fields) into the linearized PDEs, you will obtain an eigenvalue problem involving a...
View
Full Document
 Spring '11
 Staff
 Derivative, Boundary value problem, Eigenvalue, eigenvector and eigenspace, Normal mode, Boundary conditions

Click to edit the document details